Question

The value(s) of $x$ for which the value of $detA=0,$ where $A=\left[\begin{array}{ccc}x-1& 1& 1\\ 1& x-1& 1\\ 1& 1& x-1\end{array}\right]$

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

Answer: (C), (D)

$2$

$|A|=\left|\begin{array}{ccc}x-1& 1& 1\\ 1& x-1& 1\\ 1& 1& x-1\end{array}\right|$
Expanding along the first row
$|A|=(x-1)\left|\begin{array}{cc}x-1& 1\\ 1& x-1\end{array}\right|-1\left|\begin{array}{cc}1& 1\\ 1& x-1\end{array}\right|+1\left|\begin{array}{cc}1& x-1\\ 1& 1\end{array}\right|$
$=(x-1)\left(\right(x-1\left)\right(x-1)-1)-1\left(\right(x-1)-1)+1(1-(x-1\left)\right)$
$=(x-1)(x^2-2x+1-1)-(x-2)+(2-x)$
$=x(x-1)(x-2)-(x-2)-(x-2)$
$=(x-2)$[$x(x-1)-2$]
$=(x-2)(x^2-x-2)$
Now $|A|=0$
$\Rightarrow$ $(x-2)(x^2-x-2)=0$
$\Rightarrow$$(x-2)(x-2)(x+1)=0$
$\therefore x=-1$ or $2$